The dropping damage evaluation for packaging system is essential for safe transportation and storage. A dynamic model of nonlinear cubic-quintic Duffing oscillator for the suspension spring packaging system was proposed. Then, a first-order approximate solution was obtained by applying He’s variable iteration method. Based on the results, a damage evaluation equation was derived, which reveals the main controlling physical parameters for damage potential of drop to packaged products concretely. Finally, the dropping damage boundary curves and surfaces for the system were discussed. It was found that decreasing the suspension angle can improve the safe region of the system.

Let $G=(V,E)$ be a graph the atom-bond connectivity (ABC) index is defined as the sum of weights $((d_u+d_v-\mathrm{2})/d_u{d}_v{)}^{\mathrm{1}/\mathrm{2}}$ over all edges $uv$ of $G$, where $d_u$ denotes the degree of a vertex $u$ of $G$. In this paper, we determined a few structural features of the trees with minimal ABC index also we characterized the trees with $\text{d}\text{i}\text{a}[T]=\mathrm{2}$ and minimal ABC index, where $[T]$ is induced by the vertices of degree greater than 2 in $T$ and $\text{d}\text{i}\text{a}[T]$ is the diameter of $[T]$.

The author considers an impulsive boundary value problem involving the one-dimensional p-Laplacian $-{({\phi }_p (u^{\prime }))}^{\prime }=\lambda \omega \left(t\right)f\left(t,u\right),$ $t\ne t_k,$ $\mathrm{\Delta }u{|}_{t=t_k}=\mu I_k\left(t_k,u\left(t_k\right)\right),$ $\mathrm{\Delta }{u}^{\mathrm{\prime }}{|}_{t=t_k}=0,$ $k=\mathrm{1,2},\dots ,n,$ $au(0)-b{u}^{\mathrm{\prime }}(0)={\int }_0^1\mathrm{‍}g(t)u(t)dt,u^{\mathrm{\prime }}(1)=0$, where $\lambda >0$ and $\mu >0$ are two parameters. Using fixed point theories, several new and more general existence and multiplicity results are derived in terms of different values of $\lambda >0$ and $\mu >0$. The exact upper and lower bounds for these positive solutions are also given. Moreover, the approach to deal with the impulsive term is different from earlier approaches. In this paper, our results cover equations without impulsive effects and are compared with some recent results by Ding and Wang.

We investigate a class of nonperiodic fourth order differential equations with general potentials. By using variational methods and genus properties in critical point theory, we obtain that such equations possess infinitely homoclinic solutions.

We investigate a stochastic Food-Chain System $dx(t)=[r_1(t)-a_{11}(t)x-(a_{12}(t)y/(1+{\beta }_1(t)x+{\gamma }_1(t)y))]$ $xdt+{\sigma }_1(t)xd{B}_1(t)$, $dy(t)=[r_2(t)-a_{21}(t)y+(a_{22}(t)x/(1+{\beta }_1(t)x+{\gamma }_1(t)y))-(a_{23}(t)z/(1+{\beta }_2(t)y+{\gamma }_2(t)z))]$ $ydt+{\sigma }_2(t)yd{B}_2(t)$, $dz(t)=[-r_3(t)+(a_{31}(t)y/(1+{\beta }_2(t)y+{\gamma }_2(t)z))-a_{32}(t)z]zdt+{\sigma }_3(t)zd{B}_3(t)$, where $B_i(t)$, $i$ = $1,2,3,$ is a standard Brownian motion. Firstly, the existence, the uniqueness, and the positivity of the solution are proved. Secondly, the stochastically ultimate boundedness of the system is investigated. Thirdly, the boundedness of moments and upper-growth rate of the solution are obtained. Then the global attractivity of the system is discussed. Finally, the main results are illustrated by several examples.

We discuss the dynamical behavior of the stochastic delay three-specie mutualism system. We develop the technique for stochastic differential equations to deal with the asymptotic property. Using it we obtain the existence of the unique positive solution, the asymptotic properties, and the nonpersistence. Finally, we give the numerical examinations to illustrate our results.

We study the existence of multiple solutions for a fourth-order nonlinear boundary value problem. We give some new criteria for guaranteeing that the fourth-order elastic beam equation with a perturbed term has at least three solutions. The proof is based on some three critical points theorems of B. Ricceri. Furthermore, numerical simulations are also presented.

We present a survey on the existence of nontrivial solutions to impulsive differential equations by using variational methods, including solutions to boundary value problems, periodic solutions, and homoclinic solutions.

Circulant matrices have important applications in solving ordinary differential equations. In this paper, we consider circulant-type matrices with the $k$-Fibonacci and $k$-Lucas numbers. We discuss the invertibility of these circulant matrices and present the explicit determinant and inverse matrix by constructing the transformation matrices, which generalizes the results in Shen et al. (2011).

A nonautonomous plant-hare model with impulse is considered. By using the continuation theorem of coincidence degree theory, we present an easily verifiable sufficient condition on the existence of multiple periodic solutions. Though Gao et al. (2014) considered the periodic solutions of plant-hare model, such model with impulses and delay has not been studied in previous paper.

We consider the global existence of solutions to the 2D incompressible generalized liquid crystal flow. It is proved that the local solution exists globally with $\beta =0$, $\alpha \ge 2$.

Nonuniform exponential dichotomy has been investigated extensively. The essential condition of these previous results is based on the assumption that the nonlinear term satisfies $|f(t,x)|\le \mu e^{-\epsilon |t|}$. However, this condition is very restricted. There are few functions satisfying $|f(t,x)|\le \mu e^{-\epsilon |t|}$. In some sense, this assumption is not reasonable enough. More suitable assumption should be $|f(t,x)|\le \mu$. To the best of the authors' knowledge, there is no paper considering the existence and uniqueness of solution to the perturbed nonautonomous system with a relatively conservative assumption $|f(t,x)|\le \mu$. In this paper, we prove that if the nonlinear term is bounded, the perturbed nonautonomous system with nonuniform exponential dichotomy has a unique solution. The technique employed to prove Theorem 4 is the highlight of this paper.

Some new sufficient conditions to guarantee wave breaking for the modified two-component Camassa-Holm system are established.

We investigate the Cauchy problem for the modified Novikov equation. We establish blow-up criteria on the initial data to guarantee the corresponding solution blowing up in finite time.

The existence and multiplicity of homoclinic solutions for a class of first-order periodic Hamiltonian systems with spectrum point zero are obtained. The proof is based on two critical point theorems for strongly indefinite functionals. Some recent results are improved and extended.

We investigate the nonlocal boundary value problems of impulsive fractional differential equations. By Banach’s contraction mapping principle, Schaefer’s fixed point theorem, and the nonlinear alternative of Leray-Schauder type, some related new existence results are established via a new special hybrid singular type Gronwall inequality. At last, some examples are also given to illustrate the results.

A class of neural networks system with neutral delays is investigated. The existence and uniqueness of almost periodic solution for the system are obtained by using fixed point theorem; we extend some results in the references.

We develop a new split-step $\theta$ (SS$\theta$) method for stochastic age-dependent capital system with random jump magnitudes. The main aim of this paper is to investigate the convergence of the SS$\theta$ method for a class of stochastic age-dependent capital system with random jump magnitudes. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from theory.

In this paper, the peristaltic flow of a Jeffrey fluid in an asymmetric channel has been investigated. Mathematical modeling is carried out by utilizing long wavelength and low Reynolds number assumptions. Closed form expressions for the pressure gradient, pressure rise, stream function, axial velocity, and shear stress on the channel walls have been computed numerically. Effects of the Hartmann number, the ratio of relaxation to retardation times, time-mean flow, the phase angle and the gravity field on the pressure gradient, pressure rise, streamline, axial velocity, and shear stress are discussed in detail and shown graphically. The results indicate that the effect of Hartmann number, ratio of relaxation to retardation times, time-mean flow, phase angle, and gravity field are very pronounced in the peristaltic transport phenomena. Comparison was made with the results obtained in the presence and absence of magnetic field and gravity field.

The bifurcation of a nongeneric homoclinic orbit (i.e., the orbit comes from the equilibrium along the unstable manifold instead of the center manifold) connecting a nonhyperbolic equilibrium is investigated, and the nonhyperbolic equilibrium undergoes a pitchfork bifurcation. The existence (resp., nonexistence) of a homoclinic orbit and an 1-periodic orbit are established when the pitchfork bifurcation does not happen, while as the nonhyperbolic equilibrium undergoes a pitchfork bifurcation, we obtain the sufficient conditions for the existence of homoclinic orbit and two or three heteroclinic orbits, and so forth. Moreover, we explore the difference between the bifurcation of the nongeneric homoclinic orbit and the generic one.

We analyze a predator prey model with stochastic perturbation. First, we show that this system has a unique positive solution. Then, we deduce conditions that the system is persistent in time average. Furthermore, we show the conditions that there is a stationary distribution of the system which implies that the system is permanent. After that, conditions for the system going extinct in probability are established. At last, numerical simulations are carried out to support our results.

The existence and uniqueness of square-mean almost automorphic mild solution to a stochastic functional integrodifferential equation is studied. Under some appropriate assumptions, the existence and uniqueness of square-mean almost automorphic mild solution is obtained by Banach’s fixed point theorem. Particularly, based on Schauder’s fixed point theorem, the existence of square-mean almost automorphic mild solution is obtained by using the condition which is weaker than Lipschitz conditions. Finally, an example illustrating our main result is given.

We study the number and distribution of limit cycles of some planar $Z_4$-equivariant quintic near-Hamiltonian systems. By the theories of Hopf and heteroclinic bifurcation, it is proved that the perturbed system can have 24 limit cycles with some new distributions. The configurations of limit cycles obtained in this paper are new.

We discuss a third-order multipoint boundary value problem under some appropriate resonance conditions. By using the coincidence degree theory, we establish the existence result of solutions. The emphasis here is that the dimension of the linear operator is equal to two. Our results supplement other results.